Let E = 2 log3 x + log3 5
We will use the logarethim properties to simplify the expression:
We know that: a*log b = log b^a
==> E = log3 (x^2) + log3 5
Now since the bases of the logarethim are the same we could use the properties of the product of two logarethims:
The, from logarethim properties we know that:
log a + log b = log a*b
==> E = log3 (x^2)*5
= log3 5x^2
Then we can wrtie the expression 2log3 x + log3 5 as follows:
==> 2log3 x + log3 5= log3 5x^2
You will need to make use of the following rules of logarithm:
1. a.logb(c) = logb(c^a)
2. logb(a) + logb(c) = logb(a.c)
2.log3 (x) + log3 (5)
= log3 (x^2) + log3 (5)
= log3 (5x^2)
Just a warning:
Look at rule number 2 above.
It is logb(a) + logb(c) = logb(a.c)
Never end up remembering " logb(a).logb(c) = logb(a+c) " !! There is no such rule at all!
I have a little memory aid to share:
"I remember that logarithm is a transformation to make things simpler.
Logarithm transforms products into sums."
To write 2 log3 x + log3 5 as a single logarithmic expression.
By property of logarithms,
log a+logb = logab and
m loga = loga^m.
Therefore 2log3 (x) = log3 (x^2).
Therefore 2log 3 (x) +log3 (5) = log(x^2) + log3 (5).
2log 3 (x) +log3 (5) = log 3 (x^2 * 5).
2log 3 (x) +log3 (5) = log 3 (5x^2).
We see that log 3 (5x^2) is a single expression for 2log3 (x) + log3 (5).
It is easy to write the sum of logarithms as a singl term, since the bases are matching and we could apply the product rule.
But before applying the product rule, we'll have to apply the power rule of logarithms for the term 2*log 3 x:
2*log 3 x = log 3 (x^2)
We'll re-write the expression:
2 log3 x + log3 5 = log 3 (x^2) + log3 5
Now, we can apply the product rule:
log a x + log a y = log a(x*y)
We'll put a = 3:
log 3 (x^2) + log3 5 = log 3 (5*x^2)
The simplified logarithmic term is:log 3 (5*x^2)